Giving \(G\)  an orientation is just a housekeeping aid for using Kirchoff's laws.
It doesn't change the topology and is thus quite different from making \(G\)  a directed graph.
The circuit physics is the same whatever orientation is chosen.
The Signature Method (\(\Sigma \) -method) for DAEs
Our preferred SA method for DAEs is the Signature method (\(\Sigma \) -method) [1]}.
The \(\Sigma \) -method can be applied to regular nonlinear DAEs of arbitrary high order \(k\)  of the form
\(f(t,x,\dot{x},\dots ,x^{(k)})=0,\) 

with \(f:\mathbb {T}\times \mathbb {R}^n\times \dots \times \mathbb {R}^n\rightarrow \mathbb {R}^n\)  sufficiently smooth.
The \(\Sigma \) -method consists of the following steps:
Building the signature matrix \(\Sigma =[\sigma _{ij}]_{i,j=1,\dots ,n}\)  with
\(\sigma _{ij}={\left\lbrace \begin{array}{ll}\text{highest order of derivative of $x_j$ in $f_i$,} \\ -\infty \text{ if $x_j$ does not occur in $f_i$.}\end{array}\right.}\) 

Finding a highest value transversal (HVT) of \(\Sigma \) , i.e.,
a set
\({\cal T}=\lbrace (1,j_1),(2,j_2),\dots ,(n,j_n)\rbrace \) ,
where \((j_1,\dots ,j_n)\)  is a permutation of \((1,\dots ,n)\) ,
with maximal value \(\text{Val}({\cal T}):=\sum _{(i,j)\in {\cal T}}\sigma _{ij}.\)

Computing offsets \(c=[c_i]_{i=1,\dots ,n}\)  and \(d=[d_j]_{j=1,\dots ,n}\)  with \(c_i \ge 0\) , \(d_j \ge 0\)  such that
\(\text{$d_j-c_i\ge \sigma _{ij}$ for all $i,j=1,\dots ,n$, with equality for $(i,j)\in \mathcal {T}$}.\) 


Forming the system Jacobian \(\mathbf {J} = \begin{bmatrix}\mathbf {J}_{ij}\end{bmatrix}_{i,j=1}^n\)  where (the two formulae are equivalent)
\({\mathbf {J}}_{ij}:={\left\lbrace \begin{array}{ll}\displaystyle \partial f_i/\partial x_j^{(\sigma _{ij})} &\text{ if } d_j-c_i=\sigma _{ij}, \\0 &\text{ otherwise}\end{array}\right.}\quad = \quad {\left\lbrace \begin{array}{ll}\displaystyle \partial f_i/\partial x_j^{(d_j-c_i)} &\text{if $d_j-c_i\ge 0$}, \\0 &\text{otherwise}\end{array}\right.}\) 


Form the reduced derivative array \({\cal F}(t,{\cal X})=0\)  where
\({\cal X}=[x_1,\dot{x}_1,\dots ,x_1^{(d_1)}, \dots , x_n,\dot{x}_n,\dots ,x_n^{(d_n)}]^\ {\cal F}=[f_1,\dot{f}_1,\dots ,f_1^{(c_1)}, \dots , f_n,\dot{f}_n,\dots ,f_n^{(c_n)}]^\) 


Success check: if \({\cal F}(t,{\cal X})=0\) , considered locally as an algebraic system of \(M=n+\sum _{i=1}^n c_i\)  equations in \(N=n+\sum _{j=1}^n d_j\)  unknowns, has a solution \((t^*,{\cal X}^*)\in \mathbb {T}\times \mathbb {R}^N\)  where \(\mathbf {J}\)  is nonsingular, then \((t^*,{\cal X}^*)\)  is a consistent point, the method “succeeds”, and the DAE is SA-amenable.
In case of success the structural index
\(\nu _s:=\max _i c_i + (\text{$1$ if some $d_j=0$, $0$ otherwise})\) 

is \(\ge \)  the differentiation index \(\nu _d\) .
Also \(\text{Val}(\Sigma )\) , defined as the value of the highest value transversal \(\mathcal {T}\)  and also equal to \(\sum _j d_j - \sum _i c_i\) , equals the number of degrees of freedom of the system.
We have the standard solution scheme (SSS): starting with \(k = -\max d_j\) , in order of increasing stage number \(k\) ,
\(\begin{aligned}& \text{solve the equations}& &f_i^{(k+c_i)}=0&&\text{for those $i$ such that $k+c_i\ge 0$} \\& \text{for the unknowns}&&x_j^{(k+d_j)}&&\text{for those $j$ such that $k+d_j\ge 0$}\end{aligned}\biggr \rbrace \;.\) 

One interpretation of this is as a numerical scheme that first finds a consistent point and then expands the DAE solution through this point as a Taylor series; then, \(k\)  goes up to some Taylor order \(k=k_{\max }>0\) .
However the meaning used in this paper is that we go up to \(k=0\) : stages up to \(k=-1\)  are “numerical”, and a consistent point is a set of values that solve these equations; while \(k=0\)  is “symbolic”—a set of equations by which (using the Implicit Function Theorem) the highest derivatives \(x_j^{(d_j)}\)  are proved to be functions of lower derivatives, thus defining an implicit ODE.
Let \(\mathbf {J}_k\)  be the Jacobian of the equations in (REF ) at stage \(k\) .
Theorem B.1 
([1]})
For an SA-amenable DAE, each \(\mathbf {J}_k\)  is of full row rank, and for all \(k\ge 0\)  is equal to the system Jacobian \(\mathbf {J}\)  (hence is nonsingular).

Port-Hamiltonian dynamics
Port-Hamiltonian system and Dirac structure
A physical system is modelled by flows of energy within it, these flows being represented by ports: (flow, effort) pairs of variables whose product has the dimension of power.
The following paradigm is taken from [3]}.
The \(\mathpzc {F}_\bullet \)  are linear spaces and \(\mathpzc {E}_\bullet =\mathpzc {F}_\bullet ^*\)  their duals.
In our context they are some \(\mathbb {R}^d\) , so the duality inner product \(\langle \,\mathpzc {f}\hspace{-1.0pt},\,\mathpzc {e}\hspace{0.5pt}\,\rangle =\mathpzc {f}\hspace{-1.0pt}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}\) .
In general \(\mathcal {Z}\)  is a manifold, with \(T_z\mathcal {Z}\)  its tangent space at the state \(z\in \mathcal {Z}\) , and \(T_z^*\mathcal {Z}\)  the corresponding co-tangent space.
In our context \(\mathcal {Z}\)  is some \(\mathbb {R}^d\) , so \(T_z\mathcal {Z}\)  and \(T_z^*\mathcal {Z}\)  can also be regarded as \(\mathbb {R}^d\) , again with duality inner product.
A port-Hamiltonian system gives mathematical form to Figure REF . It is defined by
[(pH1)]
A state space \(\mathcal {Z}\)  and a Hamiltonian \(\mathcal {H}: \mathcal {Z}\rightarrow \mathbb {R}\) , defining energy-storage, corresponding to an energy-storing port \((\mathpzc {f}\hspace{-1.0pt}_\mathcal {S},\mathpzc {e}\hspace{0.5pt}_\mathcal {S})\in T_z\mathcal {Z}\times T_z^*\mathcal {Z}\) .

A dissipative structure
\(\mathcal {R}&\subset \mathpzc {F}_\mathcal {R}\times \mathpzc {E}_\mathcal {R}&\text{having $\langle \,\mathpzc {f}\hspace{-1.0pt},\,\mathpzc {e}\hspace{0.5pt}\,\rangle \le 0$ for all $(\mathpzc {f}\hspace{-1.0pt},\mathpzc {e}\hspace{0.5pt})\in \mathcal {R}$},\) 

corresponding to an energy-dissipating port \((\mathpzc {f}\hspace{-1.0pt}_\mathcal {R}, \mathpzc {e}\hspace{0.5pt}_\mathcal {R}) \in \mathpzc {F}_\mathcal {R}\times \mathpzc {E}_\mathcal {R}\) .

An external port \((\mathpzc {f}\hspace{-1.0pt}_\mathcal {P}, \mathpzc {e}\hspace{0.5pt}_\mathcal {P}) \in \mathpzc {F}_\mathcal {P}\times \mathpzc {E}_\mathcal {P}\) .

A Dirac structure linking (pH1)–(pH3),
\(\mathcal {D}\;&\subset \; T_z\mathcal {Z}\times T_z^*\mathcal {Z}\,\times \, \mathpzc {F}_\mathcal {R}\times \mathpzc {E}_\mathcal {R}\,\times \, \mathpzc {F}_\mathcal {P}\times \mathpzc {E}_\mathcal {P}.\) 

Definition C.1 [3]}
Consider a finite-dimensional linear space \(\mathpzc {F}\)  with dual \(\mathpzc {E}=\mathpzc {F}^*\) ,
and duality inner product \(\langle \,.,\,.\,\rangle :\mathpzc {F}\times \mathpzc {E}\rightarrow \mathbb {R}\) .
A linear subspace \(\mathcal {D}\subset \mathpzc {F}\times \mathpzc {E}\)  is a Dirac structure if \(\langle \,\mathpzc {f}\hspace{-1.0pt},\,\mathpzc {e}\hspace{0.5pt}\,\rangle = 0\) , for all \((\mathpzc {f}\hspace{-1.0pt},\mathpzc {e}\hspace{0.5pt}) \in \mathcal {D}\) , and \({\rm dim}\, \mathcal {D}= {\rm dim}\, \mathpzc {F}\) .

Here, using the natural identification \((V\times V^{\prime })^* = V^*\times V^{\prime *}\)  for linear spaces \({\hspace{0.2pt}\texttt {V}},V^{\prime }\) , the space on the right of (REF ) is viewed as the product \(\mathpzc {F}\times \mathpzc {E}\)  of
\(\mathpzc {F}&= T_z\mathcal {Z}\times \mathpzc {F}_\mathcal {R}\times \mathpzc {F}_\mathcal {P}\quad \text{and}\quad \mathpzc {E}= \mathpzc {F}^* = T_z^*\mathcal {Z}\times \mathpzc {E}_\mathcal {R}\times \mathpzc {E}_\mathcal {P},{with the inner product}\langle \,\mathpzc {f}\hspace{-1.0pt},\,\mathpzc {e}\hspace{0.5pt}\,\rangle  &= \langle \,(\mathpzc {f}\hspace{-1.0pt}_\mathcal {S},\mathpzc {f}\hspace{-1.0pt}_\mathcal {R},\mathpzc {f}\hspace{-1.0pt}_\mathcal {P}),\,(\mathpzc {e}\hspace{0.5pt}_\mathcal {S},\mathpzc {e}\hspace{0.5pt}_\mathcal {R},\mathpzc {e}\hspace{0.5pt}_\mathcal {P})\,\rangle  =\mathpzc {f}\hspace{-1.0pt}_\mathcal {S}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}_\mathcal {S} + \mathpzc {f}\hspace{-1.0pt}_\mathcal {R}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}_\mathcal {R} + \mathpzc {f}\hspace{-1.0pt}_\mathcal {P}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}_\mathcal {P} .\nonumber \) 

The \(\langle \,\mathpzc {f}\hspace{-1.0pt},\,\mathpzc {e}\hspace{0.5pt}\,\rangle = 0\)  in the definition represents the lossless routing of energy between parts of the system.
In the circuit context the Dirac structure is essentially Kirchhoff's laws for a given circuit topology.
That they define lossless routing, is Tellegen's theorem.
Port-Hamiltonian system dynamics
The dynamics of a system is specified by
\(\left(-\dot{z}(t), \frac{\partial \mathcal {H}}{\partial z}(z(t)),\; \mathpzc {f}\hspace{-1.0pt}_\mathcal {R}(t), \mathpzc {e}\hspace{0.5pt}_\mathcal {R}(t),\; \mathpzc {f}\hspace{-1.0pt}_\mathcal {P}(t), \mathpzc {e}\hspace{0.5pt}_\mathcal {P}(t) \right) &\in \mathcal {D}(z(t)), \\(\mathpzc {f}\hspace{-1.0pt}_\mathcal {R}(t), \mathpzc {e}\hspace{0.5pt}_\mathcal {R}(t)) &\in \mathcal {R}(z(t)), \nonumber \) 

for \(t\)  in some real interval.
A DAE that fits this structure—it might actually be an ODE—will be called a port-Hamiltonian DAE (pHDAE).
A key point from the physical modelling angle is that \(\mathcal {H}\)  is an algebraic function of \(z\) —the time derivative \(\dot{z}\)  must not enter—and must include the whole set of variables on which system energy depends (and preferably nothing but them).
Hence we can use the chain rule in the form
\(\dot{\mathcal {H}}(z(t)) = \frac{\partial \mathcal {H}}{\partial z}^(z(t))\;\dot{z}(t) = -\mathpzc {f}\hspace{-1.0pt}_\mathcal {S}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}_\mathcal {S},\) 

the second equality because from (REF ), \(-\dot{z}\)  is identified with \(\mathpzc {f}\hspace{-1.0pt}_\mathcal {S}\) , and \(\partial \mathcal {H}/\partial z\)  with \(\mathpzc {e}\hspace{0.5pt}_\mathcal {S}\) .
The Dirac structure enforces the power balance equation
\(\mathpzc {f}\hspace{-1.0pt}_\mathcal {S}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}_\mathcal {S} + \mathpzc {f}\hspace{-1.0pt}_\mathcal {R}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}_\mathcal {R} + \mathpzc {f}\hspace{-1.0pt}_\mathcal {P}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}_\mathcal {P} = 0\) 
and thus the DAE records energy flow: any solution of (REF ) satisfies
\(\dot{\mathcal {H}}= \mathpzc {f}\hspace{-1.0pt}_\mathcal {R}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}_\mathcal {R} + \mathpzc {f}\hspace{-1.0pt}_\mathcal {P}\mathop {\cdot }\mathpzc {e}\hspace{0.5pt}_\mathcal {P}.\) 

In particular, energy is conserved if there is no dissipation or external power flow.
Nothing is said about what variables occur in a pHDAE.
It will have the first order form \(f(t,x(t),\dot{x}(t)) = 0\) , where vector \(x(t)\) , the DAE variables, contains the state variables \(z(t)\in \mathcal {Z}\)  as a subvector.
What the remaining \(x_i\)  are, depends on the scenarios chosen to specify the dissipative structure (REF ) and the external port behaviour.

Positive definiteness and passivity results
Matrix positive definiteness
We cite without proof some results on positive definite matrices or matrix pairs that are standard, e.g. [5]}, [6]}, and prove some that are special to this paper. The facts about matrix pairs are present, but not so directly stated, in [7]}.
Definition D.1  
A matrix \(A\in \mathbb {R}^{n\times n}\)  is said to be positive definite (PD) if \(x^A̰x>0\)  for all \(x\in \mathbb {R}^n\setminus \lbrace 0\rbrace \) .
A pair of matrices \((B,B^{\prime })\) , \(B,B^{\prime }\in \mathbb {R}^{n\times n}\)  is called a positive definite matrix pair (PDMP) (resp. negative definite matrix pair (NDMP)) if for all \(y,y^{\prime }\in \mathbb {R}^n\)  not both 0, and \(By+B^{\prime }y^{\prime }=0\) , it holds that \(y^y̰^{\prime }>0\)  (resp. \(y^y̰^{\prime }<0\) ).

Note that positive definiteness of a matrix \(A\)  does not imply that \(A\)  is symmetric.
Lemma D.2  
Let \(A\in \mathbb {R}^{n\times n}\) .
[(i)]
If \(A\)  is PD, then \(A\)  is nonsingular, and \(A^{-1}\)  is PD.

\(A\)  is PD iff its symmetric part \((A+A^/2\)  is PD.

If \(A\)  is PD and \(\tilde{A}=-\tilde{A}^ is skew-symmetric, then \) A+A\( is PD.\)
Lemma D.3  
Let \((B,B^{\prime })\)  with \(B,B^{\prime }\in \mathbb {R}^{n\times n}\)  be a PDMP (resp. NDMP). Then
[(i)]
\((B^{\prime },B)\)  is a PDMP (resp. NDMP);

\(B\)  and \(B^{\prime }\)  are nonsingular and \(-B^{-1}B^{\prime }\) , \(-{B^{\prime }}^{-1}B\)  are PD (resp. \(B^{-1}B^{\prime }\) , \({B^{\prime }}^{-1}B\)  are PD);

\((PBQ,PB^{\prime }Q)\)  is a PDMP (resp. NDMP) for \(P,Q\in \mathbb {R}^{n\times n}\)  with \(P\)  nonsingular, \(Q\)  orthogonal.

\([C,C^{\prime }] = [B,B^{\prime }]\;P\)  is a PDMP (resp. NDMP), where \(P\)  is a permutation matrix as in (REF ), i.e., \((C,C^{\prime })\)  results from swapping any given set of columns of \(B\)  with the corresponding columns of \(B^{\prime }\) .
We prove the results for PDMP \((B,B^{\prime })\) .
[(i)]

Follows immediately from \({y^{\prime }}^y̰=y^y̰^{\prime }\) .

Suppose \(B\)  is singular, so there exists \(y\ne 0\)  with \(By=0\) . Setting \(y^{\prime }=0\)  gives \(By+B^{\prime }y^{\prime }=0\)  with \(y,y^{\prime }\)  not both 0, but \(y^y̰^{\prime }=0\) , contradicting the definition of a PDMP.
For the second part let \(A=-B^{-1}B^{\prime }\) . For any \(y^{\prime }\ne 0\) , set \(y=-B^{-1}B^{\prime }y^{\prime }\) .
Then \(y,y^{\prime }\)  are not both 0 and \(By+B^{\prime }y^{\prime }=0\)  so \(0<{y^{\prime }}^y̰ = -{y^{\prime }}^B̰^{-1}B^{\prime }y^{\prime } = {y^{\prime }}^A̰y^{\prime }\) . Hence \(A\)  is PD.

Let \((C,C^{\prime })=(PBQ,PB^{\prime }Q)\)  with \(P,Q\)  be as stated.
For any \(y,y^{\prime }\) , not both 0, suppose that \(0 = Cy+C^{\prime }y^{\prime } = P(Bz + B^{\prime }z^{\prime })\)  where \(z=Qy\) , \(z^{\prime }=Qy^{\prime }\) . Since \(P\)  and \(Q\)  are nonsingular, \(z,z^{\prime }\)  are not both 0 and \(Bz + B^{\prime }z^{\prime }=0\) .
As \((B,B^{\prime })\)  is a PDMP, we have \(0<z^z̰^{\prime } = y^Q̰^Q̰y^{\prime }=y^y̰^{\prime }\)  since \(Q\)  is orthogonal.
Since \(y,y^{\prime }\)  were arbitrary this shows \((C,C^{\prime })\)  that is a PDMP.

Let \(y,y^{\prime }\in \mathbb {R}^n\) , not both 0, satisfy \(Cy+Cy^{\prime }=0\) .
Defining \([w;w^{\prime }]=P_\beta [y;y^{\prime }]\) , we have \(Bw+B^{\prime }w^{\prime } = Cy+C^{\prime }y^{\prime }=0\) , with \(w,w^{\prime }\)  not both 0. So \(w^w̰^{\prime }>0\)  as \((B,B^{\prime })\)  is a PDMP.
By (REF ) it holds that \(w^w̰^{\prime }=y^y̰^{\prime }\)  (as \(w_jw_j^{\prime }\)  is replaced by \(w_j^{\prime }w_j\)  for each \(j\)  where a swap was done, but the sum stays the same).
Hence \(y^y̰^{\prime }>0\) , which proves \((C,C^{\prime })\)  is a PDMP.
Lemma D.4 
Let \(B,B^{\prime }\in \mathbb {R}^{n\times n}\) .
If \(B\)  is nonsingular and \(B^{-1}B^{\prime }\)  is PD then \((B,B^{\prime })\)  is a NDMP. In particular \(A\in \mathbb {R}^{n\times n}\)  is PD if and only if \((I,A)\)  is a NDMP.

Follows directly from the previous lemma.
Lemma D.5 
If \(M = \begin{bmatrix}M_{11} &M_{12} \\M_{21} &M_{22}\end{bmatrix}\begin{matrix}\mbox{\footnotesize $n$}_1 \\\mbox{\footnotesize $n$}_2\end{matrix}\)  is PD, then
\( P = \begin{bmatrix}M_{11} - N^M̰_{21}, &M_{12} - N^M̰_{22} \\N, &I\end{bmatrix} \) 

is nonsingular for an arbitrary \(n_2\times n_1\)  matrix \(N\) .

Take any \(w = [u;v]\)  and suppose \(Pw=0\) , so
\(0 &= (M_{11} - N^M̰_{21})u + (M_{12} - N^M̰_{22})v,\qquad 0 = Nu + v{Premultiply the first equation by u^ and substitute -Nu = v given by the second equation:}0 &= u^M̰_{11}u - u^N̰^M̰_{21}u + u^M̰_{12}v - u^N̰^M̰_{22}v \\&= u^M̰_{11}u + v^M̰_{21}u + u^M̰_{12}v + v^M̰_{22}v \qquad = w^M̰ w,\) 

so \(w=0\)  by the PD assumption, proving \(P\)  is nonsingular.
Lemma D.6 
Let \(C,C^{\prime },S\in \mathbb {R}^{n\times n}\) . If \((C,C^{\prime })\)  is a PDMP and \(S=-S^, then \) M =I S
C C'\( is nonsingular.\) 

Suppose \(z = \begin{bmatrix}x\\y\end{bmatrix}\)  with \(Mz = 0\) .
So \(x + Sy = 0\) , hence \(y^x̰ = y^-Sy)) = -y^S̰y = 0\)  since \(S\)  is skew-symmetric.
Also \(Cx + C^{\prime }y = 0\) , so as \((C,C^{\prime })\)  is a PDMP we have \(x = y = 0\) , i.e. \(z = 0\) . Hence \(M\)  is nonsingular.
Passivity results
There are various notions of passivity used in the literature. For a discussion of different definitions we refer to [8]}.
Most generally, a system is passive if it can either absorb or store energy, but cannot create energy within itself.
Passivity can be characterized by a dissipation inequality [9]}.
For port-Hamiltonian systems, passivity directly follows from the power-balance.
A port-Hamiltonian system is called passive (with respect to the port variables \(\mathpzc {f}\hspace{-1.0pt}_\mathcal {P},\mathpzc {e}\hspace{0.5pt}_\mathcal {P}\)  and storage function \(\mathcal {H}\) ) if
\(\mathcal {H}(z(t_1))-\mathcal {H}(z(t_0))\le \int _{t_0}^{t_1} \mathpzc {e}\hspace{0.5pt}_\mathcal {P}^_\mathcal {P}\, dt\) 
for all time instants \(t_0\le t_1\) .
Hence, a passive system cannot store more energy in its storage function \(\mathcal {H}\)  than it is supplied with.
We also need the concept of passivity for the dissipative structure \(\mathcal {R}\)  described by a relation \(r(\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}},\nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}})=0\) .
Definition D.7 [10]}
The resistive structure \(\mathcal {R}\)  is called passive [strictly passive] if \(\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\mathop {\cdot }\nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}} \ge 0\)  [resp. \(\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\mathop {\cdot }\nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}} > 0\) , except at the origin] for all consistent \((\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}, \nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}})\) .

Definition D.8 [11]}, [7]}
Assume that there exist a disjoint union \(\alpha \stackrel{\cdot }{\smash{\cup }}\beta ={\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\)  such that
\(\left.\left(\left[\frac{\partial r}{\partial \iota _{\alpha }},\frac{\partial r}{\partial \nu _{\beta }}\right]\right)\right|_{(\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}^*,\nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}^*)}\qquad \text{is nonsingular}\) 

at a given consistent \((\iota ^*_{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}},\nu ^*_{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}})\) .
The resistive structure is called strictly locally passive at \((\iota ^*_{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}, \nu ^*_{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}})\) if
\(D_{\cal R}:=-\begin{bmatrix}\frac{\partial r}{\partial \nu _{\alpha }}&\frac{\partial r}{\partial \iota _{\beta }}\end{bmatrix}^{-1}\cdot \begin{bmatrix}\frac{\partial r}{\partial \iota _{\alpha }}&\frac{\partial r}{\partial \nu _{\beta }}\end{bmatrix}.\) 

is positive definite at \((\iota ^*_{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}, \nu ^*_{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}})\) .
The resistive structure \(\mathcal {R}\)  is called strictly
locally passive if \(D_{\cal R}\)  is positive definite for all consistent \((\iota ^*_{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}, \nu ^*_{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}})\) .

Strict local passivity implies local passivity, the converse is not true.
Also passive components need not to be strictly locally passive as the following example shows.
Example 7 
The ideal transformer can be described by the relation
\( \begin{bmatrix}\nu _1\\\iota _2\end{bmatrix}=\begin{bmatrix}0&k\\-k&0\end{bmatrix}\begin{bmatrix}\iota _1\\\nu _2\end{bmatrix},\quad \text{where $k\in \mathbb {R}$ is the turns ratio}, \) 
see e.g. [13]}.
This resistive structure is passive, since
\(\nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\mathop {\cdot }\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}=\nu _1\iota _1+\nu _2\iota _2=k\nu _2\iota _1-k\nu _2\iota _1=0,\) 
but not strictly locally passive as (A5) is not satisfied.

From [7]} it follows that the positive definiteness of \(D_{\cal R}\)  in Definition REF  does not depend on the specific subsets \(\alpha ,\beta \) . In particular, in a strictly passive circuit one can choose arbitrarily the voltages or currents in every dissipative edge as a variable for a mixed description (REF ).
Defining
\(\bar{E}:=\frac{\partial r}{\partial \iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}}(\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}^*, \nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}^*), \text{ and } \bar{D}:=\frac{\partial r}{\partial \nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}}(\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}^*, \nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}^*)\) 
strict local passivity implies that the incremental conductance and resistance matrices
\(\bar{G}:=-\bar{E}^{-1}\bar{D} \quad \text{and}\quad \bar{R}:= -\bar{D}^{-1}\bar{E}\) 

are well-defined and positive definite (cf. [7]}).
In the linear setting (A5) reads
\(D \nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}+ E \iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}= 0\qquad \text{for some }D,E\in \mathbb {R}^{n_{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\times n_{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}}\) 

with positive definite pair \((D,E)\)  so that \(G=-E^{-1}D\)  (or equivalently \(R:= -D^{-1}E\) ) are positive definite.
Note that in the linear setting these properties are globally defined.
Within the port-Hamiltonian setting, the dissipative structure is defined by \(\mathcal {R}\subset \mathpzc {F}_\mathcal {R}\times \mathpzc {E}_\mathcal {R}\)  with \(\mathpzc {e}\hspace{0.5pt}_\mathcal {R}\mathop {\cdot }\mathpzc {f}\hspace{-1.0pt}_\mathcal {R}\le 0\) .
Non-negativity of the dissipative power flow \(\mathpzc {e}\hspace{0.5pt}_\mathcal {R}\mathop {\cdot }\mathpzc {f}\hspace{-1.0pt}_\mathcal {R}\)  respects the common convention that power flows from the “boundary” ports (i.e. ports related to \(\mathcal {P}\)  and \(\mathcal {R}\) ) into the system and from the internal network (the Dirac structure) into the energy storing elements \(\mathcal {S}\) , such that in the power balance equation all flows sum up to zero.
Here, it means the power flows from the energy-dissipating elements into the Dirac structure \(\mathcal {D}\)  (cf. [16]}).
By defining \(\nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}=\mathpzc {e}\hspace{0.5pt}_\mathcal {R}\)  and \(\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}=-\mathpzc {f}\hspace{-1.0pt}_\mathcal {R}\)  this setting corresponds to the sign convention in circuit analysis and in the definition of strict local passivity (e.g. the positive definiteness of conductance and resistance matrices) and
\(\mathpzc {e}\hspace{0.5pt}_\mathcal {R}\mathop {\cdot } \mathpzc {f}\hspace{-1.0pt}_\mathcal {R}\le 0 \quad \Longleftrightarrow \quad \iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\mathop {\cdot }\nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\ge 0. \) 
Using e.g. the voltage-controlled representation of resistors \(\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}= \bar{G} \nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\)  this gives
\(\iota _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\mathop {\cdot }\nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}} =\nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\mathop {\cdot } \bar{G} \nu _{\hspace{0.2pt}{\hspace{0.2pt}\texttt {D}}}\ge 0\quad \Longleftrightarrow \quad \bar{G}\ge 0,\) 
i.e., the resistor is passive iff \(\bar{G}\)  is positive definite.

Matlab code
Below are the Matlab functions for the circuit in Example REF  and for the Diode Clipper example generated from the CpH class, both in Model 2 form. They can be used by Matlab's [basicstyle=]ode15i solver.
flexiblefalse0truesolarized@cyansolarized@bluesolarized@redsolarized@base01solarized@base2falsefalse%@trueutf8trueφ1
numbers=left]code/daefcnExample1mm.m
flexiblefalse0truesolarized@cyansolarized@bluesolarized@redsolarized@base01solarized@base2falsefalse%@trueutf8trueφ1
numbers=left]code/daefcnDiodeClippermm.m
